A241683 Total number of 2 X 2 squares appearing in the Thue-Morse sequence logical matrices after n stages.

0, 0, 0, 2, 12, 50, 220, 882, 3612, 14450, 58140, 232562, 931612, 3726450, 14911260, 59645042, 238602012, 954408050, 3817719580, 15270878322, 61083862812, 244335451250, 977343203100, 3909372812402

Offset: 0

Author

Kival Ngaokrajang, Apr 27 2014

Keywords

• nonn

Comments

a(n) is the total number of non-isolated "1s" (consecutive 1s on 2 rows, 2 columns) that appear as 2 X 2 squares in the Thue-Morse logical matrices after n stages. See links for more details.

Links

• Kival Ngaokrajang, Illustration of initial terms
• Wikipedia, Thue-Morse sequence
• Index entries for linear recurrences with constant coefficients, signature (4, 5, -20, -4, 16).

Crossrefs

Cf. A010060.

Programs

PARI

{a0=0;print1(a0,", "); for (n=0,50, b=ceil(2*(2^n-1)/3); a=floor(b^2/2); print1(a,", "))}

Formula

a(n) = A007590(A000975(n - 1)).

Empirical g.f.: 2*x^3*(4*x^2-2*x-1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x-1)). - Colin Barker, Apr 27 2014

a(n) = (1/18) * (4^n - 3*2^n - 6*(-1)^n + (-2)^n - 2), n>0 (from g.f.). - Ralf Stephan, Apr 27 2014